Variance of the volume of random real algebraic submanifolds II

TL;DR

This paper investigates the asymptotic behavior of the variance of the volume of zero sets of random real holomorphic sections over complex projective manifolds, extending previous results to maximal codimension cases and establishing positivity of the leading constant.

Contribution

It extends previous work by covering the maximal codimension case and proving the positivity of the asymptotic variance constant.

Findings

Variance of zero set volume asymptotics as degree increases

Main theorem includes maximal codimension case

Leading constant in variance asymptotics is positive

Abstract

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ be its real locus. We study the vanishing locus $Z\_{s\_d}$ in $M$ of a random real holomorphic section $s\_d$ of $\mathcal{E} \otimes \mathcal{L}^d$, where $\mathcal{L} \to \mathcal{X}$ is an ample line bundle and $\mathcal{E} \to \mathcal{X}$ is a rank $r$ Hermitian bundle, $r \in \{1,\dots, n\}$. We establish the asymptotic of the variance of the linear statistics associated with $Z\_{s\_d}$, as $d$ goes to infinity. This asymptotic is of order $d^{r-\frac{n}{2}}$. As a special case, we get the asymptotic variance of the volume of $Z\_{s\_d}$. The present paper extends the results of [20], by the first-named author, in essentially two ways. First, our main theorem covers the case of maximal codimension ($r = n$), which was left out in [20]. And second, we show that the leading constant in our asymptotic is positive. This last result is proved by studying the Wiener--It{\=o} expansion of the linear statistics associated with the common zero set in $\mathbb{RP}^n$ of $r$ independent Kostlan--Shub--Smale polynomials.